Introduction
Construction of Lax pairs
Conclusions
A Symbolic Algorithm to Compute Lax Pairs in
Matrix Form for Nonlinear Evolution Equations
Jacob Rezac1
Willy Hereman2
1 University
of Delaware
School of Mines
IMACS Nonlinear Evolution Equations and Wave Phenomena: Computation and
Theory 2013
2 Colorado
03/26/2013
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
1
Introduction
2
Construction of Lax pairs
3
Conclusions
Collaborators: Sara Clifton, Oscar Aguilar (CSM)
Funded by NSF research award no. CCF-0830783
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
What is a matrix Lax pair?
For a PDE,
ut = f (u, ux , uxx , ...),
matrices X and T which satisfy
Φx = X Φ
and
Φt = T Φ,
for an eigenfunction Φ are called a Lax pair.
This is equivalent to the equation
Xt − Tx + [X , T ] =
˙ 0
for [X , T ] = XT − TX .
The notation =
˙ means equality on solutions of the PDE.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Example
Consider the Korteweg-de Vries (KdV) equation,
ut + αuux + uxxx = 0.
A known Lax pair is
XKdV =
TKdV =
−4λ2 +
0
1
λ − 16 αu 0
1
6 αux
1
1 2 2
αλu
+ 18
α u
3
,
+ 16 αuxx
−4λ − 13 αu
− 61 αux
.
The isospectral parameter λ (λt = 0) is necessary for non-triviality.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Example
Substituting these into the Lax equation,
0
0
(XKdV )t − (TKdV )x + [XKdV , TKdV ] =
− 16 α (ut + αuux + uxxx ) 0
0 0
=
˙
.
0 0
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Operators
We can also use operators, L and M which satisfy
Lt + [L, M] =
˙ 0.
We don’t consider these here.
M. Hickman, W. Hereman, J. Larue, Ü. Göktas, Scaling invariant
Lax pairs of nonlinear evolution equations, Applicable Analysis.
Vol. 91 (2), pp. 381-402 (2012).
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Applications
Lax pairs are useful in
Finding exact solutions of nonlinear PDEs (inverse scattering
transform)
Construction of conservation laws (see my poster tomorrow!)
“Verification” that an equation is integrable.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
How could we construct this?
Try “linear” elements for X
X1,1 + X2,1 λ + X3,1 u X1,2 + X2,2 λ + X3,2 u
XGuess =
X1,3 + X2,3 λ + X3,3 u X1,4 + X2,4 λ + X3,4 u
and up to “quadratic” elements for T , e.g.,
(1,1)
TGuess = T1,1 + T2,1 λ + T3,1 u + T4,1 ux + T5,1 λ2 + T6,1 λu
+ T7,1 u 2 + T8,1 uxx .
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
How could we construct this?
Substituting into the Lax equation,
XGuess,t − TGuess,x
L1,1 L2,2
+ [XGuess , TGuess ] =
,
L2,1 L2,2
where, e.g.,
L1,1 = −λT6,1 ux + λT4,3 X2,2 ux − λT4,2 X2,3 ux +
λT8,3 X2,2 uxx − λT8,2 X2,3 uxx + T4,3 X1,2 ux − T4,2 X1,3 ux +
T4,3 X3,2 uux − T4,2 X3,3 uux + T8,3 X1,2 uxx − T8,2 X1,3 uxx +
T8,3 X3,2 uuxx − T8,2 X3,3 uuxx − T3,1 ux − 2T7,1 uux − T4,1 uxx −
T8,1 uxxx + λ2 T6,3 X2,2 u − λ2 T6,2 X2,3 u + λ2 T5,3 X3,2 u − λ2 T5,2 X3,3 u +
λT6,2 X1,3 u + λT7,3 X2,2 u 2 + λT3,3 X2,2 u − λT7,2 X2,3 u 2 − . . .
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
How could we construct this?
. . . − λT3,2 X2,3 u + λT6,3 X3,2 u 2 + λT2,3 X3,2 u − λT6,2 X3,3 u 2 −
λT2,2 X3,3 u + T7,3 X1,2 u 2 + T3,3 X1,2 u − T7,2 X1,3 u 2 − T3,2 X1,3 u+
T7,3 X3,2 u 3 + T3,3 X3,2 u 2 + T1,3 X3,2 u − T7,2 X3,3 u 3 − T3,2 X3,3 u 2 −
T1,2 X3,3 u − αX3,1 uux − X3,1 uxxx + λ3 T5,3 X2,2 − λT6,3 X1,2 u−
T3,2 X1,3 u − λT6,2 X1,3 u + λT3,3 X2,2 u + λ3 T5,2 X2,3 + λ2 T5,3 X1,2 −
λ2 T5,2 X1,3 + λ2 T2,3 X2,2 − λ2 T2,2 X2,3 + λT2,3 X1,2 − λT2,2 X1,3 +
λT1,3 X2,2 − λT1,2 X2,3 + T1,3 X1,2 − T1,2 X1,3 .
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
How could we construct this?
This is pretty bad!
We can clean this up by matching powers. For example,
L1,1 = λ (T2,3 X1,2 − T2,2 X1,3 + T1,3 X2,2 − T1,2 X2,3 ) + other terms.
It is simpler to solve
T2,3 X1,2 − T2,2 X1,3 + T1,3 X2,2 − T1,2 X2,3 = 0,
especially with the other equations from L1,1 , L1,2 , L2,1 , and L2,2 .
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Big Idea
Use properties of PDE to simplify guess.
In particular, consider scaling invariance.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Scaling Invariance and Weight
Rescale the variables in the KdV equation to
(x, t, u) → (κ−1 x, κ−3 t, κ2 u) = (ξ, τ, ν).
(1)
Then,
ut + αuux + uxxx = 0 → κ−5 (ντ + αννξ + νξξξ ) = 0.
Call each exponent on κ in (1) the weight of a variable (denoted
W (·)). That is,
∂
∂
W
=1
W
=3
W (u) = 2.
∂x
∂t
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
J. Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
J. Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
J. Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
J. Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
J. Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
∂
∂t
=3
W (u) = 2.
Why is W (λ) = 2? Set
λ = iκ2 .
(We will implicitly take spectral parameter weight to be 1).
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
The modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
J. Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
|4iλ +
{z2iλu}




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conclusions
The modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
J. Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
|4iλ +
{z2iλu}




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conclusions
The modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
J. Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
|4iλ +
{z2iλu}




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conclusions
The modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
J. Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
|4iλ +
{z2iλu}




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conclusions
The modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
J. Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
|4iλ +
{z2iλu}




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conclusions
Construction
Assume Lax matrices have weight invariant elements, e.g.,
W (X11 ) W (X12 )
W (T11 ) W (T12 )
W (X ) =
W (T ) =
.
W (X21 ) W (X22 )
W (T21 ) W (T22 )
We substitute these weights into the Lax equation and force scale
invariance,
W (Xt ) = W (Tx ) = W ([X , T ]) ,
or,
W (X ) + W
∂
∂x
J. Rezac
∂
= W (T ) + W
∂t
= W (X ) + W (T ) .
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Solving the system of equations produced from this yields
∂
W ∂x
W (X12)
W (X ) =
∂
∂
− W (X12 ) W ∂x
2W ∂x
and
W (T ) =
∂
W
∂t
∂
W ∂t
− W (X12 ) + W
J. Rezac
∂
∂x
W
∂
∂t
+ W (X12) − W
∂
W ∂t
Computation of Lax Pairs
∂
∂x
.
Introduction
Construction of Lax pairs
Conclusions
Method of Construction
Step 1: Pick W (X12 )
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
Step 3: Substitute matrices into Lax equation.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
Step 3: Substitute matrices into Lax equation.
Step 4: Solve for undetermined coefficients.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
Step 3: Substitute matrices into Lax equation.
Step 4: Solve for undetermined coefficients.
Step 5: Substitute back into X and T .
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Example
Consider the KdV equation (α = 6) and assume (Step 1)
W (X12 ) = 0. Then, our candidate Lax pair is (Step 2)
c1 κ
c3
X =
c4 κ2 + c5 u −c1 κ
and
T =
T11 T12
T21 −T11
,
where
T11 = c6 κ3 + c7 κu + c8 ux ,
T12 = c9 κ2 + c10 u,
T21 = c11 κ4 + c12 κ2 u + c13 κux + c14 u 2 + c15 uxx .
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
We substitute this into the Lax equation (Step 3) and get, for the
(1, 1)-element,
(c11 c3 − c4 c9 ) κ4 + (c12 c3 − c10 c4 − c5 c9 ) κ2 u + (c14 c3 − c10 c5 ) u 2
+ (c13 c3 − c7 ) κux + (c15 c3 − c8 ) uxx = 0.
We break this up in powers of κ and u to get the system of
equations
c11 c3 − c4 c9 = 0,
c12 c3 − c10 c4 − c5 c9 = 0,
c14 c3 − c10 c5 = 0,
c13 c3 − c7 = 0,
c15 c3 − c8 = 0.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Solving with the equations from the other three matrix elements
(Step 4) gives many solutions. Choose (Step 5) a one:
#
"
0
− cc10
2
11
X =
c11
2 + 2c u
c
κ
0
10
11
2c10
and
T =
T11 T12
T21 −T11
,
where
T11 = ux ,
2c10
T12 = 3 −c10 κ2 + c11 u ,
c11
T21 = c10 κ4 + c11 κ2 u −
2
c11
2u 2 + uxx .
c10
Setting c10 = 4, c11 = 2i, and κ2 = −iλ gives the pair from before.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
What about W (X12 )?
Assume X and T matrices are of the form
X =
n
X
X (i) λi
and
T =
i=0
m
X
T (j) λj
j=0
where weights are chosen such that W (λ) = 1 and each element of
∂
and each element of T has weight
X has weight n = W ∂x
∂
m = W ∂t . That is,
weight n
z
X (i)
|{z}
}|
{
× |{z}
λi
weight n − i
J. Rezac
weight i
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Substitute our series forms of X and T into the Lax equation.
These can be separated by powers
of λ and split into two types of
∂
equations. Assuming W ∂x
= 1,
Algebraic constraints
[X (1) , T (m) ] =
˙ 0
and
1
X
[X (k) , T (m−k) ] =
˙ 0
k=0
Solve these algebraic equations and substitute into
Differential constraints
(0)
(0)
− T
+ [X (0) , T (0) ] =
˙ 0
X
t
x
− T (i)
x
+
1
X
[X (k) , T (m−i−k) ] =
˙ 0
k=0
These can now be solved by imposing weights.
J. Rezac
Computation of Lax Pairs
i = 1, . . . , m − 1
Introduction
Construction of Lax pairs
Conclusions
Example
Consider the mKdV equation and assume matrices of the form
X =
1
X
X (i) λi
and
i=0
T =
3
X
T (j) λj .
j=0
The algebraic constraints are
[X (1) , T (3) ] = 0,
[X (0) , T (3) ] + [X (1) , T (2) ] = 0.
To solve this, we set up
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
"
X11
X12
(i)
(i)
X21 −X11
"
T11
T12
(i)
(i)
T21 −T11
X (i) =
and
T (i) =
(i)
(i)
#
(i)
(i)
#
.
Solving, we find
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
X11 X12
X21 −X11
X =
,
where
(0)
(1)
X11 = X11 + X11 λ
(3)
X12 =
(0)
(1)
(3)
(1)
(3)
(2)
(0)
(2)
(1)
(3)
(3)
−
T11 T21 X11
(3)
(T11 )2
(2)
T11
Substituting in candidate elements gives
J. Rezac
(1)
T11 T12 X11
(1)
T21 X11 + T21 X11 + T21 X11 λ
(3)
−
T11
(3)
X21 =
(0)
T12 X11 + T12 X11 + T12 X11 λ
Computation of Lax Pairs
(3)
(3)
(T11 )2
(1)
.
Introduction
Construction of Lax pairs
Conclusions
X =
X11 X12
X21 −X11
and
T =
T11 T12
T21 −T11
.,
where
X11 = c1 λ + c2 u
1
c1 c3 c5
X12 =
c1 c7 λ + c2 c3 + c1 c6 −
u
c4
c4
c1 c5 c7
1
c1 c7 λ + c2 c7 + c1 c8 −
u .
X21 =
c4
c4
and
T11 = c4 λ3 + c5 uλ2 + (c11 u 2 + c12 ux )λ + c9 u 3 + c10 uux + c13 uxx ,
T12 = c3 λ3 + c6 uλ2 + (c16 u 2 + c17 ux )λ + c14 u 3 + c15 uux + c18 uxx
T21 = c7 λ3 + c8 uλ2 + (c21 u 2 + c22 ux )λ + c19 u 3 + c20 uux + c23 uxx .
We use these to solve the constraints with derivatives. Doing so
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
c1 λ − c22 u
2c2 u −c1 λ
T11 T12
T21 −T11
X =
and
T =
.
where
T11 = −4c13 λ3 − 2c1 λu 2 ,
1
T12 = 2c12 c2 λ2 u + c1 c2 λux + c2 u 3 + c2 uxx
2
2
T21 =
−4c1 λ2 u + 2c1 λux − 2u 3 − uxx .
c2
Setting c1 = i and c2 = 2 gives the same answer as before.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
We can only handle simple forms for Lax pairs.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problem 1: Short Pulse Equation
The Short Pulse Equation (SPE),
uxt + αu + β(u 3 )xx = 0,
with weights
∂
W
=1
∂x
W
∂
∂t
=3
W (u) = 1
has a Lax pair with X -component
λ
XSPE = 1
α λux
λux
λ
W (α) = 4
This can be constructed with the method proposed in this
talk.
At present, we wouldn’t if we didn’t already know the answer.
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
We can only handle simple forms for Lax pairs.
Gauge freedom
X̂ = GXG −1 + Gx G −1
J. Rezac
and
T̂ = GTG −1 + Gt G −1
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problem 2: Weak Lax Pairs
In, e.g., S.Y. Sakovich. On zero-curvature representations of
evolution equations. Journal of Physics A: Mathematical and
General, 28(10):28612869, 1995., the concept of a weak Lax pair is
discussed.
Burgers’ equation,
ut − uxx − uux = 0
can be expressed with the Lax pair
1
λ
λ + 12 u
X =
λ
2 −λ + 12 u
and
1
λu
T =
4 −λu + 21 u 2 + u + x
J. Rezac
λu + 12 u 2 + ux
.
−λu
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problem 2: Weak Lax Pairs
However, under the gauge transformation
"√
√
λ + √1λ
λ−
1
√
√
G=
1
λ − √λ
λ+
2
we have
√1
λ
√1
λ
#
,
1
1
1 + 12 u
X̂ =
−1
2 −1 + 12 u
and
1
T̂ =
4
u
1 2
2 u − u + ux
1 2
2u
+ u + ux
.
−u
Free of λ!
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
We can only handle simple forms for Lax pairs.
Gauge freedom
X̂ = GXG −1 + Gx G −1
J. Rezac
and
T̂ = GTG −1 + Gt G −1
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
We can only handle simple forms for Lax pairs.
Gauge freedom
X̂ = GXG −1 + Gx G −1
and
T̂ = GTG −1 + Gt G −1
Use is limited to scaling invariant PDEs
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
We can only handle simple forms for Lax pairs.
Gauge freedom
X̂ = GXG −1 + Gx G −1
and
T̂ = GTG −1 + Gt G −1
Use is limited to scaling invariant PDEs
Computational complexity
J. Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conclusions
Conclusions
Thanks!
J. Rezac
Computation of Lax Pairs